Merge branch 'master' of git.kent-dobias.com:research/first_order_singularities/paper

1 files changed, 22 insertions, 15 deletions

diff --git a/ising_scaling.tex b/ising_scaling.tex
index 32085ab..6b89cb2 100644
--- a/ising_scaling.tex
+++ b/ising_scaling.tex
@@ -142,16 +142,21 @@ to constant rescaling of $u_h$). The invariant scaling combinations that appear
 as the arguments to the universal scaling functions will come up often, and we
 will use $\xi=u_h|u_t|^{-\Delta}$ and $\eta=u_t|u_h|^{-1/\Delta}$.
 
-The analyticity of the free energy at places away from the critical point implies that the functions
-$\mathcal F_\pm$ and $\mathcal F_0$ have power-law expansions of their
-arguments about zero. For instance, when $u_t$ goes to zero for nonzero $u_h$
-there is no phase transition, and the free energy must be an analytic function
-of its arguments. It follows that $\mathcal F_0$ is analytic about zero. This
-is not the case at infinity: since $\mathcal F_0(\eta)=\eta^{D\nu}\mathcal
-F_\pm(\eta^{-1/\Delta})$ has a power-law expansion about zero, $\mathcal
-F_\pm(\xi)\sim \xi^{D\nu/\Delta}$ for large $\xi$. The nonanalyticity of
-these functions at infinite argument can therefore be understood as an artifact
-of the chosen coordinates.
+The analyticity of the free energy at places away from the critical point
+implies that the functions $\mathcal F_\pm$ and $\mathcal F_0$ have power-law
+expansions of their arguments about zero, the result of so-called Griffiths
+analyticity. For instance, when $u_t$ goes to zero for nonzero $u_h$ there is
+no phase transition, and the free energy must be an analytic function of its
+arguments. It follows that $\mathcal F_0$ is analytic about zero. This is not
+the case at infinity: since
+\begin{equation}
+  \mathcal F_\pm(\xi)
+  =\xi^{D\nu/\Delta}\mathcal F_0(\pm \xi^{-1/\Delta})+\frac1{8\pi}\log\xi^{2/\Delta}
+\end{equation}
+and $\mathcal F_0$ has a power-law expansion about zero, $\mathcal
+F_\pm$ has a series like $\xi^{D\nu/\Delta-j/\Delta}$ for $j\in\mathbb N$ at
+large $\xi$, along with logarithms. The nonanalyticity of these functions at
+infinite argument can be understood as an artifact of the chosen coordinates.
 
 For the scale of $u_t$ and $u_h$, we adopt the same convention as used by
 \cite{Fonseca_2003_Ising}. The dependence of the nonlinear scaling variables on
@@ -215,7 +220,8 @@ s=2^{1/12}e^{-1/8}A^{3/2}$, where $A$ is Glaisher's constant.
 To lowest order, this singularity is a function of the scaling invariant $\xi$
 alone. It is therefore suggestive that this should be considered a part of the
 singular free energy and moreover part of the scaling function that composes
-it. We will therefore make the ansatz that
+it. There is substantial numeric evidence for this as well. We will therefore
+make the ansatz that
 \begin{equation} \label{eq:essential.singularity}
   \operatorname{Im}\mathcal F_-(\xi+i0)=A_0\Theta(-\xi)\xi e^{-1/b|\xi|}\left[1+O(\xi)\right]
 \end{equation}
@@ -281,7 +287,7 @@ functions with different asymptotic expansions in different limits, we adopt
 another coordinate system, in terms of which a scaling function can be defined
 that has polynomial expansions in \emph{all} limits.
 
-In all dimensions, the Schofield coordinates $R$ and $\theta$ will be implicitly defined by
+The Schofield coordinates $R$ and $\theta$ are implicitly defined by
 \begin{align} \label{eq:schofield}
   u_t(R, \theta) = R(1-\theta^2)
   &&
@@ -700,13 +706,14 @@ accuracy of the fit results can be checked against the known values here.
     dat = 'data/phi_comparison.dat'
 
     set xlabel '$n$'
-    set ylabel '$|\Delta\mathcal F_0^{(m)}(0)|$'
+    set xrange [1.5:7.5]
 
+    set ylabel '$|\Delta\mathcal F_0^{(m)}(0)|$'
     set format y '$10^{%T}$'
-    set style data linespoints
     set logscale y
+
+    set style data linespoints
     set key title '\raisebox{0.5em}{$m$}' bottom left
-    set xrange [1.5:9.5]
 
     plot \
       dat using 1:2 title '0', \